Solve the inequality.Express the solution set in interval notation. $$5

Question

Solve the inequality.Express the solution set in interval notation. $$5 < \left| \frac { x - 14 } { 3 } \right| < 8$$
A) $\quad ( - 38 , - 29 ) \cup ( 1,10 )$
B) $( - 10,1 ) \cup ( 29,38 )$
C) $( - 10 , - 1 ) \cup ( 29,38 )$
D) $( - 1,29 )$
E) none of the above

Quizplus · Accepted Answer

We can start by solving the two inequalities separately:

$$5 < \frac { x - 14 } { 3 } < 8$$ and $$5 < -\frac { x - 14 } { 3 } < 8$$

Solving the first inequality, we get:

$$15<x<38$$

Solving the second inequality, we get:

$$-10<x<1$$

Now we need to check which values satisfy both inequalities. We can draw a number line and mark the values that satisfy each inequality. Then we can take the intersection of those values to get the final solution set.

|----|----|----|----|----|----|----|----|----|----|----|

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45

For the first inequality, the values that satisfy it are between 15 and 38:

|----|----|----|----|----|----|----|----|----|----|----|

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45

*............*

For the second inequality, the values that satisfy it are between -10 and 1:

|----|----|----|----|----|----|----|----|----|----|----|

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45

*...................*

The intersection of the two intervals is:

|----|----|----|----|----|----|----|----|----|----|----|

-20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45

*.....*            *............*

This gives us the solution set:

$( - 10 , - 1 ) \cup ( 29,38 )$

Solve the Inequality 5<∣x−143∣<85 < \left| \frac { x - 14 } { 3 } \right| < 85<​3x−14​​<8

Solve the Inequality $5 < \left| \frac { x - 14 } { 3 } \right| < 8$